Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Theory of Probability and Its Applications

Criterion of Weak Compactness for Families of Generalized Ensembles and Its Corollaries

Alexander Semenovich Holevo, Максим Евгеньевич Широков

Published Jan 1, 2016• 0 citations
Publisher

Abstract

One of the central results of the theory of probability measures on metric spaces is Prokhorov's theorem on weak compactness of a subset of probability measures. In the present paper this theorem is used to obtain a criterion of weak compactness for families of generalized quantum ensembles, i.e., Borel probability measures on the set of quantum states, and its applications in quantum information theory are considered.

DOI
10.1137/s0040585x97t987685

Related papers

Properties of probability measures on the set of quantum states and their applications

Максим Евгеньевич Широков

Two basic properties of the set of all probability measures on the set of quantum states and their corollaries are considered. Several applications of these properties to analysis of functional constructions widely used in quantum information theory are discussed.

arXiv:math-ph/0607019•Jul 11, 2006•6 citations

On a quantum martingale convergence theorem

Kyler Scott Bridgen Johnson, Michael J. Kozdron

It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.

arXiv:1504.03829•International Journal of Quantum Information•Oct 28, 2022•0 citations

Compactness Criterion for Families of Quantum Operations in the Strong Convergence Topology and Its Applications

Максим Евгеньевич Широков

Abstract A revised version of the compactness criterion for families of quantum operations in the strong convergence topology obtained in [1] is presented, along with a more detailed proof and the examples showing the necessity of this revision. Several criteria for the existence of limit points of a sequence of quantum operations w.r.t. the strong convergence are obtained and discussed. Applications in different areas of quantum information theory are described.

arXiv:2307.13694•Lobachevskii Journal of Mathematics•Jun 1, 2024•0 citations

Continuous ensembles and the $χ$-capacity of infinite-dimensional channels

Alexander Semenovich Holevo, Максим Евгеньевич Широков

The paper is devoted to systematic study of the $χ$-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, ``continuous'' state ensembles, defined as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory it is shown that the set of all generalized ensembles with the average (barycenter) in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be verified in the case of Bosonic Gaussian channels with constrained mean energy. The importance of the above condition is shown by considering example of a constrained channel with no optimal generalized ensemble. In the case of convex constraints a characterization of the optimal generalized ensemble is obtained extending the `` maximal distance'' property.

arXiv:quant-ph/0408176•Aug 30, 2004•11 citations

Probability Theory on the Lattice of Projections in a Hilbert Space

Kalyanapuram Rangachari Parthasarathy

In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra \( \mathcal{B} \) X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on \( \mathcal{B} \) X constitute a convex set whose extreme points are degenerate probability measures.

Texts and readings in mathematics•Jan 1, 2005•0 citations

Continuous Ensembles and the Capacity of Infinite-Dimensional Quantum Channels

Alexander Semenovich Holevo, Максим Евгеньевич Широков

This paper is devoted to the study of $\chi$-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the "maximal distance property" is obtained.

Theory of Probability and Its Applications•Jan 1, 2006•64 citations
One of the central results of the theory of probability measures on metric spaces is Prokhorov's theorem on weak compactness of a subset of probability measures. In the present paper this theorem is used to obtain a criterion of weak compactness for families of generalized quantum ensembles, i.e., Borel probability measures on the set of quantum states, and its applications in quantum information theory are considered.
It is well known in quantum information theory that a positive operator-valued measure (POVM) is the most general kind of quantum measurement. Mathematically, a quantum probability is a normalized POVM, namely, a function on certain subsets of a (locally compact and Hausdorff) sample space that satisfies the formal requirements for a probability measure and whose values are positive operators acting on a complex Hilbert space. A quantum random variable is an operator-valued function which is measurable with respect to a quantum probability. In this work, we study quantum random variables and generalize several classical limit results to the quantum setting. We prove a quantum analogue of the Lebesgue-dominated convergence theorem and use it to prove a quantum martingale convergence theorem. This quantum martingale convergence theorem is of particular interest since it exhibits nonclassical behavior; even though the limit of the martingale exists and is unique, it is not explicitly identifiable. However, we provide a partial classification of the limit through a study of the space of all quantum random variables having quantum expectation zero.
Continuous ensembles and the χχχ-capacity of infinite-dimensional channels
The paper is devoted to systematic study of the χχχ-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite, in general, ``continuous'' state ensembles, defined as probability measures on the set of all quantum states. By using compactness criteria from probability theory and operator theory it is shown that the set of all generalized ensembles with the average (barycenter) in a compact set of states is itself a compact subset of the set of all probability measures. With this in hand we give a sufficient condition for the existence of an optimal generalized ensemble for a constrained quantum channel. This condition can be verified in the case of Bosonic Gaussian channels with constrained mean energy. The importance of the above condition is shown by considering example of a constrained channel with no optimal generalized ensemble. In the case of convex constraints a characterization of the optimal generalized ensemble is obtained extending the `` maximal distance'' property.
In classical probability theory one assumes that all the events concerning a statistical experiment constitute a Boolean σ-algebra and defines a probability measure as a completely additive non-negative function which assigns the value unity for the identity element of the σ-algebra. Invariably, the σ-algebra is the Borel σ-algebra B \mathcal{B} B X (i.e., the smallest σ-algebra generated by the open subsets) of a nice topological space X. Under very general conditions it turns out that all probability measures on B \mathcal{B} B X constitute a convex set whose extreme points are degenerate probability measures.
This paper is devoted to the study of χ\chiχ-capacity, closely related to the classical capacity of infinite-dimensional quantum channels. For such channels generalized ensembles are defined as probability measures on the set of all quantum states. We establish the compactness of the set of generalized ensembles with averages in an arbitrary compact subset of states. This result enables us to obtain a sufficient condition for the existence of the optimal generalized ensemble for an infinite-dimensional channel with input constraint. This condition is shown to be fulfilled for Bosonic Gaussian channels with constrained mean energy. In the case of convex constraints, a characterization of the optimal generalized ensemble extending the "maximal distance property" is obtained.